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Simplifying 3x(7 + 2x) + -1(2 + 4x) = 22 (7 * 3x + 2x * 3x) + -1(2 + 4x) = 22 (21x + 6x2) + -1(2 + 4x) = 22 21x + 6x2 + (2 * -1 + 4x * -1) = 22 21x + 6x2 + (-2 + -4x) = 22 Reorder the terms: -2 + 21x + -4x + 6x2 = 22 Combine like terms: 21x + -4x = 17x -2 + 17x + 6x2 = 22 Solving -2 + 17x + 6x2 = 22 Solving for variable 'x'. Reorder the terms: -2 + -22 + 17x + 6x2 = 22 + -22 Combine like terms: -2 + -22 = -24 -24 + 17x + 6x2 = 22 + -22 Combine like terms: 22 + -22 = 0 -24 + 17x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4 + 2.833333333x + x2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 2.833333333x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + 2.833333333x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 2.833333333x + x2 = 0 + 4 2.833333333x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 2.833333333x + x2 = 4 The x term is 2.833333333x. Take half its coefficient (1.416666667). Square it (2.006944445) and add it to both sides. Add '2.006944445' to each side of the equation. 2.833333333x + 2.006944445 + x2 = 4 + 2.006944445 Reorder the terms: 2.006944445 + 2.833333333x + x2 = 4 + 2.006944445 Combine like terms: 4 + 2.006944445 = 6.006944445 2.006944445 + 2.833333333x + x2 = 6.006944445 Factor a perfect square on the left side: (x + 1.416666667)(x + 1.416666667) = 6.006944445 Calculate the square root of the right side: 2.450906862 Break this problem into two subproblems by setting (x + 1.416666667) equal to 2.450906862 and -2.450906862.Subproblem 1
x + 1.416666667 = 2.450906862 Simplifying x + 1.416666667 = 2.450906862 Reorder the terms: 1.416666667 + x = 2.450906862 Solving 1.416666667 + x = 2.450906862 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + x = 2.450906862 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + x = 2.450906862 + -1.416666667 x = 2.450906862 + -1.416666667 Combine like terms: 2.450906862 + -1.416666667 = 1.034240195 x = 1.034240195 Simplifying x = 1.034240195Subproblem 2
x + 1.416666667 = -2.450906862 Simplifying x + 1.416666667 = -2.450906862 Reorder the terms: 1.416666667 + x = -2.450906862 Solving 1.416666667 + x = -2.450906862 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + x = -2.450906862 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + x = -2.450906862 + -1.416666667 x = -2.450906862 + -1.416666667 Combine like terms: -2.450906862 + -1.416666667 = -3.867573529 x = -3.867573529 Simplifying x = -3.867573529Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.034240195, -3.867573529}
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